# sqrt * 4 x sqrt * 4 = 4 because sqrt * 4 = 2, so 2 x 2 = 4 but what about sqrt*1=1 ?

#### Indranil

##### Junior Member
why sqrt of 1 is 1?
As I know sqrt * 4 x sqrt * 4 = 4 because sqrt * 4 = 2, so 2 x 2 = 4 but what's about sqrt * 1?

#### Dr.Peterson

##### Elite Member
why sqrt of 1 is 1?
As I know sqrt * 4 x sqrt * 4 = 4 because sqrt * 4 = 2, so 2 x 2 = 4 but what's about sqrt * 1?
First, "sqrt" is not a number you can multiply ("*"). It is a function, so you write sqrt(x) to mean the square root of x.

The reason sqrt(4)*sqrt(4) = 4 is that the left hand side is (sqrt(4))^2, and by definition the square root is the inverse of squaring: that is, squaring undoes the square root. In general, (sqrt(x))^2 = x.

What is the square root of 1? By definition, it is the (non-negative) number whose square is 1. Since 1^2 = 1, we conclude that sqrt(1) = 1.

#### pka

##### Elite Member
why sqrt of 1 is 1?
As I know sqrt * 4 x sqrt * 4 = 4 because sqrt * 4 = 2, so 2 x 2 = 4 but what's about sqrt * 1?
To Indranil, You have been asked to use sqrt(x) which is standard function notation? Why don't you please do so?

If
$$\displaystyle a\ge 0$$ then $$\displaystyle sqrt(a)$$ is real number having the property that $$\displaystyle \left(\sqrt{a}\right)^2=a$$
So
$$\displaystyle sqrt(1)\cdot sqrt(1)=\left(\sqrt{1}\right)^2=1$$

#### HallsofIvy

##### Elite Member
If you are really asking "Why isn't it $$\displaystyle \sqrt{1}= \pm 1$$" its because $$\displaystyle \sqrt{x}$$ is a function and functions must return one value for every x. $$\displaystyle \sqrt{x}$$ is defined as "the positive number, a, such that $$\displaystyle a^2= x$$".